Electronic Journal of Differential Equations, Vol. 2003(2003), No. 29, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM
BY A LOCALLY DISTRIBUTED DAMPING
ABDELAZIZ SOUFYANE & ALI WEHBE
Abstract. We study the uniform stabilization of a Timoshenko beam by
one control force. We prove that under, one locally distributed damping, the
exponential stability for this system is assured if and only if the wave speeds
are the same.
1. Introduction
A basic linear model, developed in [19], for describing the transverse vibration
of beams is given by two coupled partial differential equations
ρwtt = (K(wx − ϕ))x ,
on (0, l) × R+
(1.1)
Iρ ϕtt (x, t) = (EIϕx )x + K(wx − ϕ).
Here, t is the time variable and x the space coordinate along the beam, the length of
which is l, in its equilibrium position. The function w is the transverse displacement
of the beam and ϕ is the rotation angle of a filament of the beam. The coefficients
ρ, Iρ , E, I and K are the mass per unit length, the polar moment of inertia of a
cross section, Young’s modulus of elasticity, the moment of inertia of a cross section
and the shear modulus respectively. The natural energy of the beam is
Z
1 l
2
2
2
2
ρ |∂t w| + Iρ |∂t ϕ| + EI |∂x ϕ| + K |∂x w − ϕ| )dx.
(1.2)
E(t) =
2 0
The aim of this paper is to study two types of stabilization of this system, the
internal and the boundary stabilization. Let us mention some results about the
stabilizability of the Timoshenko beam. The case of two boundary control force
has already been considered by Kim and Renardy [4] for the Timoshenko beam.
They proved the exponential decay of the energy E(t) by using a multiplier technique and provided numerical estimates of the eigenvalues of the operator associated
to this system, and by Lagnese & Lions [11] for the study of the exact controllability, Taylor [7] studied the boundary control of system (1.1) with variable physical
characteristics. Recently Shi & Feng [16] established the exponential decay of the
energy E(t) with locally distributed feedback (two feedback).We will first prove that
it is possible to stabilize uniformly (with respect to the initial data) this beam,
2000 Mathematics Subject Classification. 35L55, 93C05, 93C20, 93D15, 93D20, 93D30.
Key words and phrases. Timohsenko beam, strong stability, non-uniform stability,
uniform stability, Lyapunov function.
c
2003
Southwest Texas State University.
Submitted September 2, 2002. Published March 16, 2003.
1
2
A. SOUFYANE & A. WEHBE
EJDE–2003/29
which is assumed to be clamped at the two ends of (0, l), by using a unique locally
distributed feedback
ρwtt = (K(wx − ϕ))x ,
on (0, l) × R+
Iρ ϕtt = (EIϕx )x + K(wx − ϕ) − b(x)ϕt ,
w(0, t) = w(l, t) = 0;
(1.3)
ϕ(0, t) = ϕ(l, t) = 0.
where b is a positive continuous function of the space variable. Indeed, we prove
the uniform stability holds for system (1.3) if and only if the wave speeds K
ρ and
EI
are
the
same.
If
not,
the
asymptotic
stability
for
this
system
is
proved.
The
Iρ
techniques we use consist in the computation of the essential type (see the definition
in the next section) of the associated semigroups, thanks to the result of Neves et
al. [6].
Our paper is organized as follows. In section 2, we give a result of a wellposedness of the solution of the system and we study the strong asymptotic stability
EI
and the nonuniform stability of (1.3) under the assumption K
ρ 6= Iρ . In section 3,
we study the uniform stability of (1.3) under the assumption
K
ρ
=
EI
Iρ .
2. Well-posedness, asymptotic and nonuniform stability
Well-posedness. Here we outline the existence theory of the solution of the Timoshenko beam. To establish the existence and uniqueness of the solution of the
studied model we use the semigroup theory, we suppose that b ∈ C([0, l]) . The
energy space associated to system (1.3) is
H := (H01 (]0, l[) × L2 (]0, l[))2
The inner product in the energy space is defined as follows:
Z l
(Y1 , Y2 ) :=
(K(∂x u1 − w1 )(∂x u2 − w2 ) + ρv1 v2 + Iρ f1 f2 + EI(∂x w1 ∂x w2 ))dx,
0


uk
 vk 
2

where Yk = 
wk  ∈ H, k = 1, 2. In the sequel we will denote by kY k := (Y, Y ),
fk
the norm in the energy space. The system 1.3 can be written as
∂t Y (t) = LY (t),
where

w(t)
∂t w(t)

Y (t) := 
 ϕ(t)  ,
∂t ϕ(t)


0
K
∂
 ρ xx
L := 
0
K
Iρ ∂x
I
0
0
0
0
−K
ρ ∂x
0
EI
Iρ ∂xx
−
K
Iρ

0
0 

I 
−b(x)
with D(L) := ((H 2 (]0, l[) ∩ H01 (]0, l[)) × H01 (]0, l[))2 .
Proposition 2.1 ([17],[16]). The operator (L, D(L)) generates a C0 -semigroup of
contractions (eLt )t≥0 on H.
EJDE–2003/29
UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM
3
Asymptotic strong Stability and nonuniform stability. Before giving our
first result, we need to recall some results and the following definitions:
eLt is asymptotically stable if, for any Y0 ∈ H limt→∞ eLt Y0 = 0.
eLt is uniformly (or exponentially) stable if there exist ω < 0 and M > 0 such
that
keLt k ≤ M eωt t ∈ R+
For a continuous linear operator from a Banach space into itself, we define its
essential spectral radius re (L) as
n
re (L) = inf R > 0 : µ ∈ σ(L), |µ| > R implies µ is an isolated o
(2.1)
eigenvalue of finite multiplicity
It is well-known (see Gohberg-Krein [3]) that, if r(L) is the spectral radius of L
re (L) ≤ r(L);
re (L + K) = re (L) ∀K ∈ L(X), K compact
If eLt is a C0 -semigroup generated by L, then (see for instance [5]) there exist
two real numbers ω = ω(L) and ωe = ωe (L) such that
r(eLt ) = eωt ,
Lt
ωe t
re (e ) = e
(2.2)
+
∀t ∈ R .
(2.3)
ω is often called the type and ωe the essential type of the semigroup. A third real
number which plays an essential role in stability theory is the spectral abscissa s(L)
of L defined by
s(L) = sup {Re λ : λ ∈ σ(L)} .
These three real numbers are related as follows [5, Theorem 3.6.1., p. 107]:
ω(L) = max (ωe (L), s(L)) .
Clearly, the uniform stability of eLt is equivalent to ω(L) < 0. We are now ready
to state our first result.
Theorem 2.2. Assume that b ∈ C([0, l]) and
b(x) ≥ b > 0
on [b0 , b1 ] ⊂ [0, l].
(2.4)
Then:
• eLt is asymptotically stable.
EI
Lt
• If K
is non uniformly stable.
ρ 6= Iρ , then e
Before giving the proof of this theorem we need to recall the following result.
Theorem 2.3 (Benchimol [2]). Let L be a maximal linear operator in a complex
Hilbert space H and assume that:
(a) L has a compact resolvent.
(b) L does not have purely imaginary eingenvalues.
Then eLt is strongly stable.
Theorem 2.2. To prove strong stability of eLt , it remains to verify the properties
(a) and (b) of theorem 2.3.
Property (a) follows at once from the compactness of the imbedding D(L) in H,
a consequence of Rellich’s theorem.
4
A. SOUFYANE & A. WEHBE
EJDE–2003/29
Suppose the conclusion of (b) is false, then L have a purely imaginary eingenvalues iω. Let U1 the eigenvectors associated to iω. Using the definition of L it
follows that LU1 = iωU1 if and only if
K∂xx u − K∂x v = −ρω 2 u,
EI.∂xx v + K∂x u − Kv − ib(x)ωv = −Iρ ω 2 v,
u = v = 0.
0,l
(2.5)
0,l
We multiply the second equation of this system by v to obtain
Z l
Z l
(EI|∂x v|2 − K∂x u.v + (K − Iρ ω 2 )|v|2 )dx = 0 and
ωb(x)|v|2 dx = 0.
0
0
Then v = 0, on [b0 , b1 ], and
EI.∂xx v + K∂x u − Kv = −Iρ ω 2 v,
which implies ∂x u = 0 on ]b0 , b1 [. From
K∂xx u − K∂x v = −ρω 2 u,
it follows that u = 0 on ]b0 , b1 [.
Now, it is trivial to see that u = v = 0 is the solution of the system
K∂xx u − K∂x v = −ρω 2 u,
EI.∂xx v + K∂x u − Kv = −Iρ ω 2 v,
(2.6)
v = u = 0 on ]b0 , b1 [,
u = v = 0.
0,l
0,l
This complete the proof of the strong stability of (eLt )t≥0 .
The proof of the nonuniform stability for eLt when K
ρ 6=
Lt
computation of the essential type of (e )t≥0 . Let


0 0
0
0
0 0
0
0

D=
0 0
0
0
K
0 0 − Iρ 0
EI
Iρ
is based on the
This operator is compact in the energy space H. Then re (L) = re (L − D). Now we
calculate the essential spectral radius of L1 := L − D , for this reason we transform
the equations by introducing the following variables:
s
s
K
K
p=−
∂x u + ∂t u, q =
∂x u + ∂t u,
ρ
ρ
s
s
EI
EI
ϕ=−
∂x v + ∂t v, ψ =
∂x v + ∂t v.
Iρ
Iρ
which are solutions of the system
 
 
 
p
p
p
ϕ
ϕ
ϕ

 
 
∂t 
 q  + K∂x  q  + C  q  = 0,
ψ
ψ
ψ
(2.7)
EJDE–2003/29
UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM
5
with the boundary conditions
(p + q)(0, t) = (ϕ + ψ)(0, t) = (p + q)(l, t) = (ϕ + ψ)(l, t) = 0,
where
p
K/ρ p 0
0
 0
EI/I
ρ
p0
K=
 0
0
− K/ρ
0
0
0
q

Iρ
K
0
− 2ρ
0
EI
√
√
b(x)
 Kρ
− 2Iρ
− 2IKρ

ρ
q
C =  2Iρ
Iρ
 0
K
−
0
√
2ρ
EI
√
b(x)
Kρ
−
− 2IKρ
2Iρ
2Iρ
ρ

0

0
,

p0
− EI/Iρ
q 
Iρ
EI 

− b(x)
2Iρ 
q

Iρ 
K
2ρ
EI 
− b(x)
2Iρ
K
2ρ
EI
L1 t
When K
)=
ρ 6= Iρ , we can apply a result in [6, p 324] which implies that re (e
L2 t
re (e ) = exp(αt) where


p

0
0
0
0
K/ρ p 0
0
0
0 − b(x) 0
 0

0 
EI/Iρ
0

2Iρ
 ∂x + 
p0
L2 = 



 0
0
0
0
0 
0
0
− K/ρ

p
b(x)
0
0
0 − 2Iρ
0
0
0
− EI/Iρ
and α = sup{Re(λ) : λ ∈ σ(L2 )}. To compute α we solve the system
s
s
K
K
∂x p = λp,
∂x q = −λq,
ρ
ρ
s
s
b(x)
b(x)
EI
EI
∂x ϕ = (λ +
)ϕ,
∂x ψ = −(λ +
)ψ,
Iρ
2Iρ
Iρ
2Iρ
with the boundary conditions
(p + q)(0, t) = (ϕ + ψ)(0, t) = (p + q)(l, t) = (ϕ + ψ)(l, t) = 0.
Then we have
r
p(x) = c1 exp(λ
ρ
x),
K
r
r
q(x) = −c1 exp(−λ
ρ
x),
K
Z x
Iρ
b(y)
(λx +
dy)),
ϕ(x) = c2 exp(
EI
0 2Iρ
r
Z x
Iρ
b(y)
ψ(x) = −c2 exp(−
(λx +
dy)),
EI
0 2Iρ
where c1 and c2 are in R\{0}. Using the boundary conditions, we see that one of
the following two equations must hold for non-trivial solutions,
r
r
ρ
ρ
exp(λ
l) − exp(−λ
l) = 0,
K
K
r
r
Z l
Z l
Iρ
b(y)
Iρ
b(y)
exp(
(λl +
dy)) − exp(−
(λl +
dy) = 0,
EI
2I
EI
ρ
0
0 2Iρ
6
A. SOUFYANE & A. WEHBE
EJDE–2003/29
these imply that
1
Re(λ) = 0 or Re(λ) = −
l
l
Z
(
0
b(x)
)dx,
2Iρ
and thus
α = sup{Re(λ), λ ∈ σ(L2 )} = 0.
Then re (eL2 t ) = 1. Using the property r(eLt ) ≥ re (eLt ) we deduce, the non uniform
EI
stability of the semigroup associated to the Timoshenko beam if K
ρ 6= Iρ .
3. Uniform stability
In this section our main result is the following theorem.
Theorem 3.1. Assuming that b(x) satisfies (2.4), eLt is exponentially stable if
K
EI
ρ = Iρ .
The proof of the uniform stability is based on the construction of a Lyapunov
function φ(Y (t)) satisfying the following inequalities:
i) There exist two positives constants, d0 , d1 such that
2
d0 kY k ≤ φ(Y ) ≤ d1 kY k
2
∀ Y D(L)
(3.1)
ii) There exist a positive constant d2 such that
2
∂t φ(Y (t)) ≤ −d2 kY (t)k
(3.2)
To construct this function we will use the multiplicative technique.
Lemma 3.2. If φ(Y (t)) satisfies (3.1), (3.2), then there exist m and e a positive
constant such that
kY (t)k2 ≤ m exp(−et)kY (0)k2
2
Proof. It follows from (3.1) that − kY (t)k ≤ − d11 φ(Y (t)). Using (3.2) then we
have
d2
∂t φ(Y (t)) ≤ − φ(Y (t))
d1
thus
d2
φ(Y (t)) ≤ exp(− t)φ(Y ) .
d1
Using (3.1), we have kY (t)k2 ≤ dd10 exp(− dd21 t)kY k2
Lemma 3.3. For any positive constant ε1 and scalar functions h, c such that:
h(0) > 0, h(l) < 0, hx < 0 on [b0 , b1 ], hx > 0 on [0, l]\[b0 , b1 ], and cxx < 0 on [0, l],
we have the following statements:
i)
Z l
Z
Z
−EI l
Iρ l
EI
∂t
Iρ ∂t ϕ.h(x)ϕx dx =
hx (ϕx )2 dx −
hx (∂t ϕ)2 dx +
[h.(ϕx )2 ]l0
2
2 0
2
0
0
Z l
Z l
−
(b(x)∂t ϕ).h(x)ϕx dx + K
(wx − ϕ).hϕx dx,
0
0
ii)
Z
l
Z
Iρ ∂t ϕ.c(x)ϕdx = − EI
∂t
0
0
l
EI
c(ϕx ) dx +
2
2
Z
l
2
Z
cxx (ϕ) dx + Iρ
0
0
l
c(∂t ϕ)2 dx.
EJDE–2003/29
UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM
l
Z
Z
l
(wx − ϕ).cϕdx −
+K
7
(b(x)∂t ϕ).c(x)ϕdx.
0
0
iii)
l
Z
ρ∂t w.ε1 (−∂xx )−1 ∂x (h(x)ϕx + c(x)ϕ)dx
∂t
0
l
Z
l
Z
=K
ε1 h(x)ϕx + ε1 c(x)ϕdx.
Z
0
0
l
K(wx − ϕ).(ε1 h(x)ϕx + ε1 c(x)ϕ)dx
ϕdx +
0
l
Z
ρ∂t w.((−∂xx )−1 ∂x (ε1 h(x)∂t ϕx + ε1 c(x)∂t ϕ))dx,
+
0
where (−∂xx )−1 is taken in the sense of Dirichlet boundary conditions.
Proof. i) Multiplying the second equation in 1.3) by h(x)ϕx , we get
Z l
Z l
Z l
∂t
Iρ ∂t ϕ.h(x)ϕx dx =
Iρ ∂tt ϕ.h(x)ϕx dx +
Iρ ∂t ϕ.h(x)∂t ϕx dx
0
0
0
l
Z
Z
l
(EIϕxx + K(wx − ϕ) − b(x)∂t ϕ).h(x)ϕx dx +
=
0
l
Z
Iρ ∂t ϕ.h(x)∂t ϕx dx
0
=
(−
0
EI
Iρ
(ϕx )2 hx − (∂t ϕ)2 hx )dx
2
2
l
Z
(K(wx − ϕ) − b(x)∂t ϕ).h(x)ϕx dx +
+
0
EI
[h.(ϕx )2 ]l0
2
ii) Multiplying the second equation of (1.3) by c(x)ϕ, we get
Z l
∂t
Iρ ∂t ϕ.c(x)ϕdx
0
l
Z
l
Z
=
Iρ ∂tt ϕ.c(x)ϕdx +
Iρ ∂t ϕ.c(x)∂t ϕdx
0
0
l
Z
Z
(EIϕxx + K(wx − ϕ) − b(x)∂t ϕ).c(x)ϕdx +
=
0
l
Iρ ∂t ϕ.c(x)∂t ϕdx
0
Z
= −EI
l
c(ϕx )2 dx +
0
EI
2
Z
l
cxx (ϕ)2 dx + K
0
l
Z
−
l
(wx − ϕ).cϕdx
0
Z
(b(x)∂t ϕ).c(x)ϕdx + Iρ
0
Z
l
c(∂t ϕ)2 dx
0
iii) Multiplying the first equation of (1.3) by
f = (−∂xx )−1 ∂x (ε1 h(x)ϕx + ε1 c(x)ϕ), f (0) = f (l) = 0,
we obtain
Z l
∂t
(ρ∂t w.f )dx
0
l
Z
l
Z
K∂x (wx − ϕ).f dx +
=
0
ρ∂t w.(∂t f )dx
0
Z
=−
l
Z
K(wx − ϕ).fx dx +
0
l
ρ∂t w.(∂t f )dx
0
8
A. SOUFYANE & A. WEHBE
EJDE–2003/29
l
Z
K(wx − ϕ).∂x (−∂xx )−1 ∂x (ε1 h(x)ϕx + ε1 c(x)ϕ)dx +
=−
0
Z
l
ρ∂t w.(∂t f )dx
0
Note that −∂x (−∂xx )−1 ∂x v = v −
Z l
∂t
(ρ∂t w.f )dx
1
l
Rl
0
v dx. Then
0
l
Z
K(wx − ϕ).(ε1 h(x)ϕx + ε1 c(x)ϕ)dx
=
0
1
l
−
Z
l
Z
0
l
Z
K(wx − ϕ)dx +
ε1 h(x)ϕx + ε1 c(x)ϕdx.
0
l
ρ∂t w.(∂t f )dx
0
l
Z
K l
=
K(wx − ϕ).(ε1 h(x)ϕx + ε1 c(x)ϕ)dx +
ε1 h(x)ϕx
l 0
0
Z l
Z l
+ ε1 c(x)ϕdx.
ϕdx −
ρ∂t w.((−∂xx )−1 ∂x (ε1 h(x)∂t ϕx + ε1 c(x)∂t ϕ))dx.
Z
0
0
Now we introduce the function
Z l
Z l
Z l
φ1 (Y (t)) := E(t) + ε1
Iρ ∂t ϕ.h(x)ϕx dx + ε1
Iρ ∂t ϕ.c(x)ϕdx −
(ρ∂t w.f )dx
0
0
0
Lemma 3.4. For positive constants ai (i = 1, 4) and
α = |(hx ∂x (−∂xx )−1 − h)|L2 (0,l) ,
β = |∂x (−∂xx )−1 |L2 (0,l)
we have
|∂t φ1 (Y (t))|
Z l
Iρ
a1 b2
a2 b2
ε1 α ε1 βc2
≤
(−b(x) − ε1 hx + ε1 Iρ c + ε1
+ ε1
+
+
)(∂t ϕ)2 dx
2
2
2
2a
2a
3
4
0
Z l
h2
EI
EI
+
(−ε1
hx − ε1 EIc + ε1
)(ϕx )2 dx + ε1
[h.(ϕx )2 ]l0
2
2a
2
1
0
Z l
EI
c2
+
(ε1
cxx + ε1
)(ϕ)2 dx
2
2a2
0
Z l
Z l
+ K
(ε1 h(x)ϕx + ε1 c(x)ϕ)dx.
ϕdx
0
+(
0
ε1 a3 α ε1 a4 β
+
)
2
2
Z
l
(ρ∂t w)2 dx.
0
Proof. Note that
∂t φ1 (Y (t))
Z
=∂t E(t) + ε1 ∂t
Z
l
0
Z
Z
0
Z
0
l
EI
[h.(ϕx )2 ]l0 )
2
l
(ρ∂t w.f )dx
0
Z l
EI
Iρ
b(x)(∂t ϕ)2 dx + ε1 ( (−
(ϕx )2 hx − (∂t ϕ)2 hx )dx
2
2
0
(K(wx − ϕ) − b(x)∂t ϕ).h(x)ϕx dx +
+
l
Iρ ∂t ϕ.c(x)ϕdx − ∂t
Iρ ∂t ϕ.h(x)ϕx dx + ε1 ∂t
0
=−
l
EJDE–2003/29
UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM
9
l
Z
Z l
EI l
cxx (ϕ)2 dx + K
(wx − ϕ).cϕdx
2 0
0
0
Z l
Z l
−
(b(x)∂t ϕ).c(x)ϕdx + Iρ
c(∂t ϕ)2 dx)
Z
+ ε1 (−EI
c(ϕx )2 dx +
0
Z
0
l
−
K(wx − ϕ).(ε1 h(x)ϕx + ε1 c(x)ϕ)dx
0
−
K
l
Z l
−
Z
l
Z
(ε1 h(x)ϕx + ε1 c(x)ϕ)dx.
0
l
ϕdx
0
ρ∂t w.((−∂xx )−1 ∂x (ε1 h(x)∂t ϕx + ε1 c(x)∂t ϕ))dx
0
l
Z l
EI
Iρ
2
(−ε1
hx − ε1 EIc)(ϕx )2 dx
= (−b(x) − ε1 hx + ε1 Iρ c)(∂t ϕ) dx +
2
2
0
0
Z l
Z l
Z l
− ε1
(b(x)∂t ϕ).h(x)ϕx dx − K
(ε1 h(x)ϕx + ε1 c(x)ϕ)dx.
ϕdx
Z
0
+ ε1
Z
−
EI
EI
[h.(ϕx )2 ]l0 + ε1
2
2
0
Z
0
l
cxx (ϕ)2 dx − ε1
0
Z
l
(b(x)∂t ϕ).c(x)ϕdx
0
l
ρ∂t w.((−∂xx )−1 ∂x (ε1 h(x)∂t ϕx + ε1 c(x)∂t ϕ))dx.
0
Z l
Iρ
EI
2
= (−b(x) − ε1 hx + ε2 Iρ c)(∂t ϕ) dx +
(−ε1
hx − ε1 EIc)(ϕx )2 dx
2
2
0
0
Z l
Z
Z l
K l
− ε1
(b(x)∂t ϕ).h(x)ϕx dx −
(ε1 h(x)ϕx + ε1 c(x)ϕ)dx.
ϕdx
l 0
0
0
Z
Z l
EI
EI l
+ ε1
[h.(ϕx )2 ]l0 + ε1
cxx (ϕ)2 dx − ε1
(b(x)∂t ϕ).c(x)ϕdx
2
2 0
0
Z l
Z l
−1
−
ρε1 (hx ∂x (−∂xx ) − h)∂t w.∂t ϕdx +
ρε1 ∂x (−∂xx )−1 ∂t w.c(x)∂t ϕdx.
Z
l
0
0
Using Young’s inequality we obtain
Z
Z l
Z l
l
1
(b(x)∂t ϕ).h(x)ϕx dx ≤ a1
(b(x)∂t ϕ)2 dx +
(h(x)ϕx )2 dx,
2 0
2a1 0
0
Z l
Z l
Z
l
1
(b(x)∂t ϕ).c(x)ϕdx ≤ a2
(b(x)∂t ϕ)2 dx +
(c(x)ϕ)2 dx,
2
2a
2
0
0
0
Z l
Z
Z
l
ε
a
α
ε1 α l
1
3
−1
2
ρε1 (hx ∂x (−∂xx ) − h)∂t w.∂t ϕdx ≤
(ρ∂t w) dx +
(∂t ϕ)2 dx ,
2
2a3 0
0
0
Z
Z
Z
l
ε1 a4 β l
ε1 β l
−1
2
ρε1 ∂x (−∂xx ) ∂t w.c(x)∂t ϕdx ≤
(ρ∂t w) dx +
(c(x)∂t ϕ)2 dx.
2
2a4 0
0
0
Then
|∂t φ1 (Y (t))|
Z l
Iρ
a1 b2
a2 b2
ε1 α ε1 βc2
≤
(−b(x) − ε1 hx + ε1 Iρ c + ε1
+ ε1
+
+
)(∂t ϕ)2 dx
2
2
2
2a3
2a4
0
10
A. SOUFYANE & A. WEHBE
EJDE–2003/29
l
EI
h2
hx − ε1 EIc + ε1
)(ϕx )2 dx
2
2a
1
0
Z l
EI
EI
c2
+ ε1
[h.(ϕx )2 ]l0 +
(ε1
cxx + ε1
)(ϕ)2 dx
2
2
2a2
0
Z
Z l
Z l
K l
ε1 a3 α ε1 a4 β
+
(ε1 h(x)ϕx + ε1 c(x)ϕ)dx.
ϕdx + (
+
) (ρ∂t w)2 dx.
l 0
2
2
0
0
Z
+
(−ε1
Lemma 3.5. For scalar positive constants ai (i = 5, 6), and scalar functions k, d
such that k(0) > 0, k(l) < 0, kx < 0 on [b0 , b1 ], kx > 0 on [0, l]\[b0 , b1 ], and d > 0
on [0, l], we have: i)
Z l
Z l
Z l
K
a5
1
∂t
ρ∂t w.k(x)wx dx ≤
(− kx + K )(wx )2 dx + K
(k(x)ϕx )2 dx
2
2
2a
5 0
0
0
Z l
ρ
K
−
kx (∂t w)2 dx + [k.(wx )2 ]l0 ,
2
0 2
ii)
l
Z
−∂t
l
Z
Z
l
K(wx − ϕ).∂x (d(x)w)dx −
ρ∂t w.d(x)wdx =
0
0
ρd(x)(∂t w)2 dx,
0
and iii)
∂t
l
Z
Z
(∂t ϕ.(ϕ − wx ) − ∂t w.ϕx )dx
0
l
≤
0
K
a6
(− +
)(ϕ − wx )2 dx +
Iρ
2Iρ
Z
l
(1 +
0
K
Iρ b2
)(∂t ϕ)2 dx + [ϕx .wx ]l0 .
2a6
ρ
Proof. i) Multiplying the first equation in (1.3) by k(x)wx , we obtain
Z l
∂t
ρ∂t w.k(x)wx dx
0
l
Z
=
l
Z
ρ∂tt w.k(x)wx dx +
ρ∂t w.k(x)∂t wx dx
0
0
l
Z
Z
K∂x (wx − ϕ).k(x)wx dx +
=
0
l
ρ∂t w.k(x)∂t wx dx
0
Z
=−
0
l
K
kx (wx )2 dx −
2
l
Z
Z
Kϕx .k(x)wx dx −
0
0
l
ρ
K
kx (∂t w)2 dx + [k.(wx )2 ]l0
2
2
Using Young’s inequality we obtain
Z l
Z
Z l
a5 l
1
ϕx .k(x)wx dx ≤
(wx )2 dx +
(k(x)ϕx )2 dx.
2
2a
5
0
0
0
Then
Z
l
Z
Iρ ∂t ϕ.h(x)ϕx dx ≤
∂t
0
l
(−
0
Z
−
0
l
K
a5
1
kx + K )(wx )2 dx + K
2
2
2a5
ρ
K
kx (∂t w)2 dx + [k.(wx )2 ]l0 .
2
2
Z
0
l
(k(x)ϕx )2 dx
EJDE–2003/29
UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM
11
ii) Multiplying the first equation in (1.3) by d(x)w, we obtain
Z l
Z l
Z l
−∂t
ρ∂t w.d(x)wdx = −
ρ∂tt w.d(x)wdx −
ρ∂t w.d(x)∂t wdx
0
0
0
l
Z
Z
=−
l
K∂x (wx − ϕ).d(x)wdx −
0
0
Z l
l
Z
K(wx − ϕ).∂x (d(x)w)dx −
=
0
ρd(x)(∂t w)2 dx
ρd(x)(∂t w)2 dx
0
iii) Multiplying the first equation in (1.3) by ϕx and the second equation of system
(1.3) by ϕ − wx , we obtain
Z l
∂t
(∂t ϕ.(ϕ − wx ) − ∂t w.ϕx )dx
0
l
Z
l
Z
∂tt ϕ.(ϕ − wx )dx +
=
∂t ϕ.(∂t ϕ − ∂t wx )dx
0
0
l
Z
−
l
Z
∂tt w.ϕx dx −
0
l
Z
∂t w∂t ϕx dx
0
EI
ϕxx .(ϕ − wx )dx −
Iρ
=
0
Z
K
(ϕ − wx )2 dx −
Iρ
0
l
Z
l
Z
l
∂t ϕ.(∂t ϕ − ∂t wx )dx −
+
0
0
K
ρ
Z
l
Z
l
b(x)
∂t ϕ.(ϕ − wx )dx
Iρ
0
Z l
K
∂x (wx − ϕ).ϕx dx −
∂t w∂t ϕx dx.
ρ
0
EI
Iρ ,
Using the fact that
=
we have
Z l
∂t
(∂t ϕ.(ϕ − wx ) − ∂t w.ϕx )dx
0
Z
l
=−
0
K
(ϕ − wx )2 dx −
Iρ
l
Z
0
b(x)
∂t ϕ.(ϕ − wx )dx +
Iρ
(∂t ϕ)2 dx −
0
K
[ϕx .wx ]l0 .
ρ
Using Young’s inequality we have
Z
Z l
Z l
l b(x)
Iρ
a6
∂t ϕ.(ϕ − wx )dx ≤
(b(x)∂t ϕ)2 dx +
(ϕ − wx )2 dx
I
2a
2I
ρ
6
ρ
0
0
0
Therefore,
Z l
∂t
(∂t ϕ.(ϕ − wx ) − ∂t w.ϕx )dx
Z
0
l
≤
0
K
a6
(− +
)(ϕ − wx )2 dx +
Iρ
2Iρ
Z
l
(1 +
0
K
Iρ b2
)(∂t ϕ)2 dx + [ϕx .wx ]l0 .
2a6
ρ
Now, we define the Lyapunov function associated to this problem.
Z l
φ(Y (t)) =φ1 (Y (t)) + ε2
(∂t ϕ.(ϕ − wx ) − ∂t w.ϕx )dx
0
Z
l
ρ∂t w.(k(x)wx − d(x)w)dx.
+ ε3
0
(3.3)
12
A. SOUFYANE & A. WEHBE
EJDE–2003/29
Lemma 3.6. For εi (i = 1, 2, 3) sufficiently small and c, h, k, d satisfying
hx (x)
+ c(x) < 0 ∀ x ∈]0, l[\]b0 , b1 [
2
hx (x)
+ c(x) > 0 ∀ x ∈]0, l[
2
kx (x)
+ d(x) > 0 ∀ x ∈]0, l[,
2
the function φ(Y (t)) satisfies (3.1) and (3.2).
−
Proof. Note that
Z
l
(∂t ϕ.(ϕ − wx ) − ∂t w.ϕx )dx
∂t φ(Y (t)) =∂t φ1 (Y (t)) + ε2 ∂t
0
Z
l
ρ∂t w.(k(x)wx − d(x)w)dx.
+ ε3 ∂t
0
Using the Lemmas 3.3 and 3.4, we obtain
|∂t φ(Y (t))|
Z l
Iρ
a1 b2
a2 b2
ε1 α
≤
(−b(x) − ε1 hx + ε1 Iρ c(x) + ε1
+ ε1
+
2
2
2
2a3
0
2
2
Iρ b
ε1 βc
+
+ ε2 (1 +
))(∂t ϕ)2 dx
2a4
2a6
Z l
(h(x))2
ε3
EI
hx − ε1 EIc + ε1
+K
(k(x))2 )(ϕx )2 dx
+
(−ε1
2
2a
2a
1
5
0
Z l
Z l
EI
(c(x))2
K
a6
+
(ε1
cxx + ε1
)(ϕ)2 dx + ε2
(− +
)(ϕ − wx )2 dx
2
2a2
Iρ
2Iρ
0
0
Z
Z
Z l
l
K l
+ ε3 K(wx − ϕ).∂x (d(x)w)dx + (ε1 h(x)ϕx + ε1 c(x)ϕ)dx.
ϕdx
l 0
0
0
K
EI
Kε
3
l
2 l
2 l
+ ε2 | [ϕx .wx ]0 | + ε1
[h.(ϕx ) ]0 +
[k.(wx ) ]0
ρ
2
2
Z l
K
a5
+ ε3
(− kx + K )(wx )2 dx
2
2
0
Z l
ε3 ρ
ε1 a3 α ε1 a4 β 2
+
(−
kx − ε3 ρd(x) + (
+
)ρ )(∂t w)2 dx .
2
2
2
0
Using Cauchy Shwartz’s inequality and Young’s inequality, we obtain
Z
Z
Z l
l
Ka7 l
K
2
K(wx − ϕ).(d(x)wx )dx ≤
(wx − ϕ) dx +
(d(x)wx )2 dx
2
2a7 0
0
0
where a7 > 0,
Z
Z
Z
2
l
Ka7 l
Kd c0 l
2
K(wx − ϕ).dx wdx ≤
(wx − ϕ) dx +
(wx )2 dx
2
2a7
0
0
0
where d = max(dx ),
Z
Z l
K l
(ε1 h(x)ϕx + ε1 c(x)ϕ)dx.
ϕdx
l 0
0
EJDE–2003/29
UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM
≤
K l 2 ε1
l 2
Z
l
(h(x)ϕx )2 dx +
0
l
Z
(
0
13
Kl2 ε1 (2 + c2 (x)) 2
)ϕ dx.
2
Then
|∂t φ(Y (t))| ≤
l
Iρ
a1 b2
a2 b2
ε1 α
hx + ε1 Iρ c(x) + ε1
+ ε1
+
2
2
2
2a3
0
2
2
ε1 βc
Iρ b
+
+ ε2 (1 +
))(∂t ϕ)2 dx
2a4
2a6
Z l
ε3
EI
(h(x))2
hx − ε1 EIc + ε1
+K
(k(x))2
+
(−ε1
2
2a
2a
1
5
0
l 2 ε1
Kl2 ε1 (2 + c2 )
(h(x))2 + c0 (
))(ϕx )2 dx
+K
2l
2
Z l
EI
(c(x))2
+
(ε1
cxx + ε1
)(ϕ)2 dx
2
2a
2
0
Z l
ε2 K
ε2 a6
+
(−
+
+ ε3 Ka7 )(ϕ − wx )2 dx
Iρ
2Iρ
0
EI
K
EI
K
+ (ε2
+ ε1
h(l))(ϕx (l))2 + (ε2
− ε1
h(0))(ϕx (0))2
2ρ
2
2ρ
2
Kε3
K
Kε3
K
+
k(l))(wx (l))2 + (ε2
−
k(0))(wx (0))2
+ (ε2
2ρ
2
2ρ
2
Z l
2
K
a5
Kd c0
Kd2
+ ε3
(− kx + K
+
+
)(wx )2 dx
2
2
2a7
2a7
0
Z l
ε3 ρ
ε1 a3 α ε1 a4 β 2
+
(−
kx − ε3 ρd(x) + (
+
)ρ )(∂t w)2 dx,
2
2
2
0
Z
(−b(x) − ε1
where c0 is the Poincarré constant and c = max(c(x)) on [0, l], then we choose εi
(i = 1, 2, 3) sufficiently small and we choose the constant ai (i = 1 . . . 7) such that
a2 b2
Iρ b2
a1 b2
+ ε1
+ ε2
< 0,
2
2
2a6
Iρ
ε1 α ε1 βc2
−ε1 hx + ε1 Iρ c(x) +
+
+ ε2 < 0,
2
2a3
2a4
EI
(h(x))2
ε3
l 2 ε1
−ε1
hx (x) − ε1 EIc(x) + ε1
+K
(k(x))2 + K
(h(x))2 < 0,
2
2a1
2a5
2l
ε2 K
ε2 a6
ε3 Ka7
−
+
+
< 0,
Iρ
2Iρ
2
ε3 ρ
ε1 a3 α ε1 a4 β 2
−
kx − ε3 ρd(x) + (
+
)ρ < 0,
2
2
2
2
EI
(c(x))
cxx +
< 0,
2
2a2
−b(x) + ε1
2
K
a5
Kd c0
Kd2
kx + K
+
+
< 0,
2
2
2a7
2a7
ρ
ρ
ε2 < min(−ε1 EI h(l), ε1 EI h(0), −ρε3 k(l), ρε3 k(0))
K
K
−
14
A. SOUFYANE & A. WEHBE
EJDE–2003/29
Then φ(Y (t)) satisfies (3.2). The inequality (3.1) follows from the Cauchy Schwartz
inequality. Thus the system is exponentially stable.
Acknowledgements. The authors wish to thank the anonymous referee for his
or her valuable suggestions. A. Soufyane would like to thank Dr. A. Benabdallah
for her help.
References
[1] Afilal M. and Ammar Khodja F., Stability of coupled second order equations.
Prépublications de l’Equipe de Mathématiques de Besançon, 1997.
[2] Benchimol C.D, A note on week Stabilizability of contractions semigroups, SIAM J. Control
and Optimization, 16, pp 373.-379, 1978.
[3] Gohberg and Krein M., Introduction to the Theory of Linear non Selfadjoint Operators,
Translations of Math. Monographs, Vol 18, Amer. Math. Soc. Providence, R. I., 1969.
[4] Kim J. U. and Renardy Y., Boundary control of the Timoshenko beam, SIAM J. Control
and Optimization, 25, 6 pp 1417-1429, 1987.
[5] van Neerven J., The Asymptotic Behaviour of a Semigroup of Linear Operators, Birkhäuser
Verlag, 1996.
[6] Neves A. F., Ribeiro H. S. and Lopes O., On the spectrum of Evolution operators generated
by hyperbolic systems, J. Functional Analysis, 67, 3, 1986.
[7] Taylor S. W., A smoothing property of a hyperbolic system and boundary controllability
Control of partial differential equations (Jacksonville, FL, 1998). J. Comput. Appl. Math.
114(2000), no.1, 23-40.
[8] Kapitonov B. V., Uniform stabilization and exact controllability for a class of coupled
hyperbolic systems. Comp. Appl. Math., Vol 15, No 3, pp 199-212, 1996.
[9] Komornik V., Exact Controllabillity And Stabilization. The Multiplier Method, Masson,
1994.
[10] Lagnese J. E., Boundary Stabilization of Thin Plates. SIAM, 1989.
[11] Lagnese J. E. and Lions J. L., Modelling Analysis and Control of Thin Plates, Masson,
1988.
[12] Morgul O., Boundary Control of a Timoshenko Beam Attached to a rigid Body: Planar
motion. Internat. J. Control 54, No. 4, pp 763-791, 1991.
[13] Morgul O., Dynamic Boundary Control of a Timoshenko Beam. Automatica J. IFAC 28,
No. 6, pp 1255-1260, 1992.
[14] Pazy A., Semigroup of linear Operators and Applications to Partial Differential Equations,
Applied Mathematical Sciences, Vol. 44, Springer -Verlag, 1983.
[15] Russell D. L., A General Framework for the study of Indirect Damping Mechanisms in
Elastic Systems. Journal of Mathematical Analysis And Applications 173, 339-358, 1993.
[16] Shi D. H. and Feng D. X., Exponential decay of Timoshenko Beam with locally distributed
feedback. Proceeding of the 99’IFAC World Congress, Beijing, Vol F.
[17] Soufyane A., Stabilisation de la poutre deTimoshenko.C.R. Acad.Sci. Paris,Ser.I, 328, pp
731-734, 1999.
[18] Soufyane A., Uniform stability of coupled second order equations. Electron. J. Diff. Eqns.,
Vol. 2001(2001),No. 25, pp 1-10.
[19] Timoshenko S., On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philisophical magazine 41, pp 744-746, 1921.
[20] Zhuang Y. and Baras J. S., Distributed control of a Timoshenko beam. In Proceeding of
the Third International Conference on Adaptative Structures, San Diago, California, Nov.
9-11, 1992.
Abdelaziz Soufyane
Math & Comp Sciences department, UAE University, PO Box 17551 Al Ain, UAE
E-mail address: Soufyane.Abdelaziz@uaeu.ac.ae
Ali Wehbe
Universite Libanaise Faculte des Sciences, Section 1, Hadath, Beyrouth, Lebanon
E-mail address:
ali wehbe@yahoo.fr